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The Symplectic Sum Formula for Gromov-Witten Invariants

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The Symplectic Sum Formula for Gromov-Witten Invariants Annals of Mathematics, 159 (2004), 935–1025 The symplectic sum formula for Gromov-Witten invariants By Eleny-Nicoleta Ionel and Thomas H. Parker* Abstract In the symplectic category there is a ‘connect sum’ operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten in- variants of a symplectic sum Z = X#Y in terms of the relative GW invariants of X and Y . Several applications to enumerative geometry are given. Gromov-Witten invariants are counts of holomorphic maps into symplectic manifolds. To define them on a symplectic manifold (X, ω) one introduces an almost complex structure J compatible with the symplectic form ω and forms the moduli space of J-holomorphic maps from complex curves into X and the compactified moduli space, called the space of stable maps. One then imposes constraints on the stable maps, requiring the domain to have a certain form and the image to pass through fixed homology cycles in X. When the correct number of constraints is imposed there are only finitely many maps satisfying the constraints; the (oriented) count of these is the corresponding GW invariant. For complex algebraic manifolds these symplectic invariants can also be defined by algebraic geometry, and in important cases the invariants are the same as the curve counts that are the subject of classical enumerative algebraic geometry. In the past decade the foundations for this theory were laid and the in- variants were used to solve several long-outstanding problems. The focus now is on finding effective ways of computing the invariants. One useful technique is the method of ‘splitting the domain’, in which one localizes the invariant to the set of maps whose domain curves have two irreducible components with the constraints distributed between them. This produces recursion relations relating the desired GW invariant to invariants with lower degree or genus. This paper establishes a general formula describing the behavior of GW in- variants under the analogous operation of ‘splitting the target’. Because we *The research of both authors was partially supported by the NSF. The first author was also supported by a Sloan Research Fellowship. 936 ELENY-NICOLETA IONEL AND THOMAS H. PARKER work in the context of symplectic manifolds the natural splitting of the target is the one associated with the symplectic cut operation and its inverse, the symplectic sum. The symplectic sum is defined by gluing along codimension two submani- folds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n−2)- submanifold V . Given a similar pair (Y,V ) with a symplectic identification between the two copies of V and a complex anti-linear isomorphism between the normal bundles NX V and NY V of V in X and in Y , we can form the sym- plectic sum Z = X#V Y . Our main theorem is a ‘Symplectic Sum Formula’ which expresses the GW invariants of the sum Z in terms of the relative GW invariants of (X, V ) and (Y,V ) introduced in [IP4]. The symplectic sum is perhaps more naturally seen not as a single manifold but as a family depending on a ‘squeezing parameter’. In Section 2 we construct a family Z → D over the disk whose fibers Zλ are smooth and symplectic for λ = 0 and whose central fiber Z0 is the singular manifold X ∪V Y .Ina neighborhood of V , the total space Z is NX V ⊕ NY V and the fiber Zλ is defined by the equation xy = λ where x and y are coordinates in the normal ∼ ∗ bundles NX V and NY V = (NX V ) . The fibration Z → D extends away from V as the disjoint union of X × D and Y × D, and the entire fibration Z can be given an almost K¨ahler structure. The smooth fibers Zλ, depicted in Figure 1, are symplectically isotopic to one another; each is a model of the symplectic sum. The overall strategy for proving the symplectic sum formula is to relate the holomorphic maps into Z0, which are simply maps into X and Y which match along V , with the holomorphic maps into Zλ for λ close to zero. This strategy involves two parts: limits and gluing. For the limiting process we consider sequences of stable maps into the family Zλ of symplectic sums as the ‘neck size’ λ → 0. In particular, these are stable maps into a compact region of the almost K¨ahler manifold Z, so that the compactness theorem for stable maps applies, giving limit maps into the singular manifold Z0 obtained by identifying X and Y along V . Along the way several things become apparent. First, the limit maps are holomorphic only if the almost complex struc- tures on X and Y match along V . To ensure this we impose the “V -compat- ibility” condition (1.10) on the almost complex structure. But there is a price to pay for that specialization. In the symplectic theory of Gromov-Witten invariants we are free to perturb (J, ν) without changing the invariant; this freedom can be used to ensure that intersections are transverse. After impos- ing the V -compatibility condition, we can no longer perturb (J, ν) along V at will, and hence we cannot assume that the limit curves are transverse to V . In fact, the images of the components of the limit maps meet V at points with well-defined multiplicities and, worse, some components may be mapped entirely into V . THE SYMPLECTIC SUM FORMULA 937 To count stable maps into Z0 we look first on the X side and ignore the maps which have marked points, double points, or whole components mapped into V . The remaining “V -regular” maps form a moduli space which is the MV union of components s (X) labeled by the multiplicities s =(s1,...,s)of MV the intersection points with V . We showed in [IP4] how these spaces s (X) can be compactified and used to define relative Gromov-Witten invariants V GWX . The definitions are briefly reviewed in Section 1. XY Figure 1. Limiting curves in Zλ = X#λY as λ → 0. Second, as Figure 1 illustrates, connected curves in Zλ can limit to curves whose restrictions to X and Y are not connected. For that reason the GW invariant, which counts stable curves from a connected domain, is not the ap- propriate invariant for expressing a sum formula. Instead one should work with the ‘Gromov-Taubes’ invariant GT, which counts stable maps from domains that need not be connected. Thus we seek a formula of the general form V ∗ V (0.1) GTX GTY =GTZ where ∗ is some operation that adds up the ways curves on the X and Y sides match and are identified with curves in Zλ. That necessarily involves keeping track of the multiplicities s and the homology classes. It also involves accounting for the limit maps with nontrivial components in V ; such curves are not counted by the relative invariant and hence do not contribute to the left side of (0.1). We postpone this issue by first analyzing limits of curves which are δ-flat in the sense of Definition 3.1. A more precise analysis reveals a third complication: the squeezing process is not injective. In Section 5 we again consider a sequence of stable maps fn into Zλ as λ → 0, this time focusing on their behavior near V , where the fn do not uniformly converge. We form renormalized maps fˆn and prove that both the domains and the images of the renormalized maps converge. The images converge nicely according to the leading order term of their Taylor expansions, but the domains converge only after we fix certain roots of unity. These roots of unity are apparent as soon as one writes down formulas. Each stable map f : C → Z0 decomposes into a pair of maps f1 : C1 → X and f2 : C2 → Y which agree at the nodes of C = C1 ∪ C2. For a specific example, suppose that f is such a map that intersects V at a single point p 938 ELENY-NICOLETA IONEL AND THOMAS H. PARKER with multiplicity three. Then we can choose local coordinates z on C1 and w on C2 centered at the node, and coordinates x on X and y on Y so that f1 3 3 and f2 have expansions x(z)=az + ··· and y(w)=bw + ···. To find maps into Zλ near f, we smooth the domain C to the curve Cµ given locally near the node by zw = µ and require that the image of the smoothed map lie in Zλ, which is locally the locus of xy = λ. In fact, the leading terms in the formulas for f1 and f2 define a map F : Cµ → Zλ whenever λ = xy = az3 · bw3 = ab (zw)3 = ab µ3 and conversely every family of smooth maps which limit to f satisfies this equation in the limit (cf. Lemma 5.3). Thus λ determines the domain Cµ up to a cube root of unity. Consequently, this particular f is, at least a priori, close to three smooth maps into Zλ — a ‘cluster’ of order three. Other maps f into Z0 have larger associated clusters (the order of the cluster is the product of the multiplicities with which f intersects V ). The maps within a cluster have the same leading order formula but have different smoothings of the domain. As λ → 0 the cluster coalesces, limiting to the single map f. This clustering phenomenon greatly complicates the analysis. To distin- guish the curves within each cluster and make the analysis uniform in λ as λ → 0, it is necessary to use ‘rescaled’ norms and distances which magnify distances as the clusters form.
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